Pith Number

pith:WT4HYX7D

pith:2026:WT4HYX7DAYRCYAR7PGZWVNSQKB

not attested not anchored not stored refs pending

Non-isotopic surfaces in $T^4\#(S^2\times S^2)$: an example

Jianfeng Lin, Yue Wu

Infinitely many embedded tori in T^4 # (S^2 × S^2) with a common geometric dual are homotopic, diffeomorphic but not isotopic even after stabilizations.

arxiv:2604.05805 v2 · 2026-04-07 · math.GT

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\pithnumber{WT4HYX7DAYRCYAR7PGZWVNSQKB}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We prove that there exist infinitely many embedded tori with a common geometric dual in T^4#(S^2×S^2) that are homotopic, diffeomorphic, but not isotopic to each other, even after arbitrary many external stabilizations.

C2weakest assumption

The isotopy classes of these surfaces are distinguished by homotopy classes of the 2-handles (relative to the boundary) in the complement of the image of the 0- and 1-handles, assuming this homotopy invariant is preserved under isotopy and that the Norman trick with non-homotopic arcs produces the claimed embedded surfaces.

C3one line summary

Infinitely many non-isotopic embedded tori with a common geometric dual exist in T^4#(S^2×S^2), built via the Norman trick on a fixed immersed surface using non-homotopic tubing arcs and distinguished by homotopy classes of 2-handles in the complement.

Receipt and verification

First computed	2026-06-25T01:17:52.909271Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b4f87c5fe306222c023f79b36ab650507ad47d3aa88ba8a4678550b376ad77d3

Aliases

arxiv: 2604.05805 · arxiv_version: 2604.05805v2 · doi: 10.48550/arxiv.2604.05805 · pith_short_12: WT4HYX7DAYRC · pith_short_16: WT4HYX7DAYRCYAR7 · pith_short_8: WT4HYX7D

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WT4HYX7DAYRCYAR7PGZWVNSQKB \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: b4f87c5fe306222c023f79b36ab650507ad47d3aa88ba8a4678550b376ad77d3

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "0cd6d975dc08077c7b179e0e1006b99cb1affc452cc59d0a51e10716e1dad5d6",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "primary_cat": "math.GT",
    "submitted_at": "2026-04-07T12:42:01Z",
    "title_canon_sha256": "ced5782f44eb4964eb436114d2a1a9c2a3c4c368d503b4ba50e21d028a5968bb"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.05805",
    "kind": "arxiv",
    "version": 2
  }
}